What, in the name of Science, are the following:

Sideways Shuffle kick: I've heard this mentioned a few times, by TKD practioners, I can basically visualise somebody shaking their leg and jumping up and down with a moronic expression on their face. What exactly is it?

could the first be a shuffle sidekick?http://www.youtube.com/watch?v=fEPJSGEtk5g
If you did that sidekick over and over while sliding forward on your support leg?
It was hella hard finding sidekick videos, cuz everyone was whoring their lame Sidekick cellphones.

If by shuffle sideways kick you mean a step sidekick then it's the one in the first video - you move your rear leg forward, just behind your lead leg, and with that momentum you shoot your lead leg into a sidekick.

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